Module: Finite Automata ( fa ) ¶

static dotDrawTransition(st1, label, st2, sep='\n')[source]¶

Draw a transition in dot format

Parameters

st1 (str) – departing state
sym (str) – label
st2 (str) – arriving state
sep (str) – separator

Return type

dotFormat(size='20,20', filename=None, direction='LR', strict=False, maxlblsz=6, sep='\n') → str[source]¶

A dot representation

Parameters

direction (str) – direction of drawing - “LR” or “RL”
size (str) – size of image
filename (str) – output file name
sep (str) – line separator
maxlblsz (int) – max size of labels before getting removed
strict (bool) – use limitations of label sizes

Returns

the dot representation

Return type

New in version 0.9.6.

Changed in version 1.2.1.

eliminateDeadName()[source]¶

Eliminates dead state name (common.DeadName) renaming the state

Returns: self
Return type: DFA

Attention

works inplace

New in version 1.2.

equivalentP(other)[source]¶

Test equivalence between automata

Parameters: other (FA) – the other automata
Return type: bool

New in version 0.9.6.

abstract evalSymbol(stil, sym)[source]¶: Evaluation of a single symbol

finalP(state: int) → bool[source]¶

Tests if a state is final

Parameters: state (int) – state index.
Returns: is the state final?
Return type: bool

finalsP(states: set) → bool[source]¶

Tests if al the states in a set are final

Parameters: states (set) – set of state indexes.
Returns: are all the states final?
Return type: bool

New in version 1.0.

hasStateIndexP(st: int) → bool[source]¶

Checks if a state index pertains to an FA

Parameters: st (int) – index of the state.
Return type: bool

images(sti, c)[source]¶

The set of images of a state by a symbol

Parameters

sti (int) – state
c (object) – symbol

Return type

iterable

indexList(lstn)[source]¶

Converts a list of stateNames into a set of stateIndexes.

Parameters: lstn (list) – list of names
Returns: the list of state indexes
Return type: set
Raises: DFAstateUnknown – if a state name is unknown

initialP(state: int) → bool[source]¶

Tests if a state is initial

Parameters: state – state index
Returns: is the state initial?
Return type: bool

initialSet()[source]¶

The set of initial states

Returns: set of States.
Return type: set

inputS(i)[source]¶

Input labels coming out of state i

Parameters: i (int) – state
Returns: set of input labels
Return type: set of str

New in version 1.0.

noBlankNames()[source]¶

Eliminates blank names

Returns: self
Return type: FA

Attention

in place transformation

plus()[source]¶: Plus of a FA (star without the adding of epsilon)

New in version 0.9.6.

renameState(st, name)[source]¶

Rename a given state.

Parameters

st (int) – state index.
name (object) – name.

Returns

self.

Return type

FA

Note

Deals gracefully both with int and str names in the case of name collision.

Attention

the object is modified in place

renameStates(name_list=None)[source]¶

Renames all states using a new list of names.

Parameters: name_list (list) – list of new names.
Returns: self.
Return type: FA
Raises: DFAerror – if provided list is too short.

Note

If no list of names is given, state indexes are used.

Attention

the object is modified in place

reversal()[source]¶

Returns a NFA that recognizes the reversal of the language

Returns: NFA recognizing reversal language
Return type: NFA

same_nullability(s1: int, s2: int) → bool[source]¶

Tests if this two states have the same nullability

Parameters

s1 (int) – state index.
s2 (int) – state index.

Returns

have the states the same nullability?

Return type

setFinal(statelist)[source]¶

Sets the final states of the FA

Parameters: statelist (int|list|set) – a list (or set) of final states indexes.

Caution

Erases any previous definition of the final state set.

setInitial(stateindex)[source]¶

Sets the initial state of a FA

Parameters: stateindex (int) – index of the initial state.

setSigma(symbol_set)[source]¶

Defines the alphabet for the FA.

Parameters: symbol_set (list|set) – alphabet symbols

stateAlphabet(sti: int) → list[source]¶

Active alphabet for this state

Parameters: sti (int) – state
Return type: list

stateIndex(name, auto_create=False)[source]¶

Index of given state name.

Parameters

name (object) – name of the state.
auto_create (bool, optional) – flag to create state if not already done.

Returns

state index

Return type

Raises

DFAstateUnknown – if the state name is unknown and autoCreate==False

Note

Replaces stateName

Note

If the state name is not known and flag is set creates it on the fly

New in version 1.0.

stateName(name, auto_create=False)[source]¶

Index of given state name.

Parameters

name (object) – name of the state
auto_create (bool, optional) – flag to create state if not already done

Returns

state index

Return type

Raises

DFAstateUnknown – if the state name is unknown and autoCreate==False

Deprecated since version 1.0: Use: stateIndex() instead

Deprecated since version 1.0: Use the stateIndex() function instead

abstract succintTransitions()[source]¶: Collapsed transitions

union(other)[source]¶

A simple literate invocation of __or__

Parameters: other (FA) – right-hand operand.
Returns: Union of self and other.
Return type: FA

words(stringo=True)[source]¶

Lexicographical word generator

Parameters: stringo (bool, optional) – are words strings? Default is True.
Yields: Word – the next word generated.

Attention

Does not generate the empty word

New in version 0.9.8.

class SemiDFA[source]¶

Class of automata without initial or final states

Variables

States (list) – set of states.
sigma (set) – alphabet set.
delta (dict) – the transition function.

dotDrawState(sti: int, sep='\n') → str[source]¶

Dot representation of a state

Parameters

sti (int) – state index.
sep (str, optional) – separator.

Returns

line to add to the dot file.

Return type

static dotDrawTransition(st1: str, lbl1: str, st2, sep='\n') → str[source]¶

Draw a transition in dot format

Parameters

st1 (str) – departing state.
lbl1 (str) – label.
st2 (str) – arriving state.
sep (str, optional) – separator.

Returns

line to add to the dot file.

Return type

dotFormat(size='20,20', filename=None, direction='LR', strict=False, maxlblsz=6, sep='\n') → str[source]¶

A dot representation

Parameters

direction (str) – direction of drawing - “LR” or “RL”
size (str) – size of image
filename (str) – Name of the output file
sep (str) – line separator
maxlblsz (int) – max size of labels before getting removed
strict (bool) – use limitations of label sizes

Returns

the dot representation

Return type

New in version 0.9.6.

Changed in version 1.2.1.

OFA¶

class OFA[source]¶

Base class for one-way automata

Variables

States (list) – set of states.
sigma (set) – alphabet set.
Initial (int) – the initial state index.
Final (set) – set of final states indexes.
delta (dict) – the transition function.

acyclicP(strict=True)[source]¶

Checks if the FA is acyclic

Parameters: strict (bool) – if not True loops are allowed

Returns: True if the FA is acyclic: bool: True if the FA is acyclic

abstract addTransition(st1, sym, st2)[source]¶

Add transition

Parameters

st1 (int) – departing state
sym (str) – label
st2 (int) – arriving state

abstract deleteStates(del_states)[source]¶

To be implemented below

Parameters: del_states (list) – states to be deleted

dotDrawTransition(st1, label, st2, sep='\n')[source]¶

Draw a transition in dot format

Parameters

st1 (str) – departing state
label (str) – symbol
st2 (str) – arriving state
sep (str) – separator

Return type

dump()[source]¶

Returns a python representation of the object

Returns: the python representation (Tags,States,sigma,delta,Initial,Final)
Return type: tuple

dup()[source]¶

Duplicate OFA

Returns: duplicate object
Return type: OFA

eliminateStout(st)[source]¶

Eliminate all transitions outgoing from a given state

Parameters: st (int) – the state index to loose all outgoing transitions

Attention

performs in place alteration of the automata

New in version 0.9.6.

emptyP()[source]¶

Tests if the automaton accepts an empty language

Return type: bool

New in version 1.0.

abstract evalSymbol(stil, sym)[source]¶: Eval symbol

abstract finalCompP(s)[source]¶

To be implemented below

Parameters: s – state
Return type: list

abstract initialComp()[source]¶

Initial component

Return type: list

minimalBrzozowski()[source]¶

Constructs the equivalent minimal DFA using Brzozowski’s algorithm

Returns: equivalent minimal DFA
Return type: DFA

minimalBrzozowskiP()[source]¶

Tests if the FA is minimal using Brzozowski’s algorithm

Return type: bool

abstract stateChildren(_state, _strict=None)[source]¶

To be implemented below

Parameters

_state (state) –
_strict (int) – state id queried

Return type

list

abstract succintTransitions()[source]¶: Collapsed transitions

topoSort()[source]¶

Topological order for the FA

Returns: List of state indexes
Return type: list

Note

self loops are taken in consideration

trim()[source]¶

Removes the states that do not lead to a final state, or, inclusively, that can’t be reached from the initial state. Only useful states remain.

Return type: FA

Attention

in place transformation

trimP()[source]¶

Tests if the FA is trim: initially connected and co-accessible

Return type: bool

abstract uniqueRepr()[source]¶: Abstract method

abstract usefulStates()[source]¶: To be implemented below

DFA¶

class DFA[source]¶

Class for Deterministic Finite Automata.

Variables

States (list) – set of states.
sigma (set) – alphabet set.
Initial (int) – the initial state index.
Final (set) – set of final states indexes.
delta (dict) – the transition function.
delta_inv (dict) – possible inverse transition map
i (bool) – is inverse map computed?

Delta(state, symbol)[source]¶

Evaluates the action of a symbol over a state

Parameters

state (int) – state index
symbol – symbol

Returns

the action of symbol over state

Return type

HKeqP(other, strict=True)[source]¶

Tests the DFA’s equivalence using Hopcroft and Karp’s state equivalence algorithm

Parameters

other –
strict –

Returns

bool

See also

J. E. Hopcroft and r. M. Karp.A Linear Algorithm for Testing Equivalence of Finite Automata.TR 71–114. U. California. 1971

Attention

The automaton must be complete.

MyhillNerodePartition()[source]¶: Myhill-Nerode partition, Moore’s way

New in version 1.3.5.

Attention

No state should be named with DeadName. This states is removed from the obtained partition.

See also

F.Bassino, J.David and C.Nicaud, On the Average Complexity of Moores’s State Minimization Algorihm, Symposium on Theoretical Aspects of Computer Science

aEquiv()[source]¶

Computes almost equivalence, used by hyperMinimial

Returns: partition of states
Return type: dict

Note

may be optimized to avoid dupped

addTransition(sti1, sym, sti2)[source]¶

Adds a new transition from sti1 to sti2 consuming symbol sym.

Parameters

sti1 (int) – state index of departure
sti2 (int) – state index of arrival
sym (str) – symbol consumed

Raises

DFAnotNFA – if one tries to add a non deterministic transition

compat(s1, s2, data)[source]¶

Tests compatibility between two states.

Parameters

data –
s1 (int) – state index
s2 (int) – state index

Return type

complete(dead='DeaD')[source]¶

Transforms the automata into a complete one. If sigma is empty nothing is done.

Parameters: dead (str) – dead state name
Returns: the complete FA
Return type: DFA

Note

Adds a dead state (if necessary) so that any word can be processed with the automata. The new state is named dead, so this name should never be used for other purposes.

Attention

The object is modified in place.

Changed in version 1.0.

completeMinimal()[source]¶

Completes a DFA assuming it is a minimal and avoiding de destruction of its minimality If the automaton is not complete, all the non final states are checked to see if tey are not already a dead state. Only in the negative case a new (dead) state is added to the automaton.

Return type: DFA

Attention

The object is modified in place. If the alphabet is empty nothing is done

completeP()[source]¶

Checks if it is a complete FA (if delta is total)

Returns: bool

completeProduct(other)[source]¶

Product structure

Parameters: other – the other DFA

computeKernel()[source]¶

The Kernel of a ICDFA is the set of states that accept a non finite language.

Returns: triple (comp, center , mark) where comp are the strongly connected components, center the set of center states and mark the kernel states
Return type: tuple

concat(fa2, strict=False)[source]¶

Concatenation of two DFAs. If DFAs are not complete, they are completed.

Parameters

strict (bool) – should alphabets be checked?
fa2 (DFA) – the second DFA

Returns

the result of the concatenation

Return type

Raises

DFAdifferentSigma – if alphabet are not equal

concatI(fa2, strict=False)[source]¶

Concatenation of two DFAs.

Parameters

fa2 (DFA) – the second DFA
strict (bool) – should alphabets be checked?

Returns

the result of the concatenation

Return type

Raises

DFAdifferentSigma – if alphabet are not equal

New in version 0.9.5.

Note

this is to be used with non complete DFAs

delTransition(sti1, sym, sti2, _no_check=False)[source]¶

Remove a transition if existing and perform cleanup on the transition function’s internal data structure.

Parameters

_no_check (bool) – use unsecure code?
sti1 (int) – state index of departure
sti2 (int) – state index of arrival
sym (str) – symbol consumed

Note

Unused alphabet symbols will be discarded from sigma.

deleteStates(del_states)[source]¶

Delete given iterable collection of states from the automaton.

Parameters: del_states – collection of int representing states

Note

in-place action

Note

delta function will always be rebuilt, regardless of whether the states list to remove is a suffix, or a sublist, of the automaton’s states list.

static deterministicP()[source]¶

Yes it is deterministic!

Return type: bool

dist()[source]¶

Evaluate the distinguishability language for a DFA

Return type: DFA

See also

Cezar Câmpeanu, Nelma Moreira, Rogério Reis: The distinguishability operation on regular languages. NCMA 2014: 85-100

New in version 0.9.8.

distMin()[source]¶

Evaluates the list of minimal words that distinguish each pair of states

Returns: set of minimal distinguishing words
Return type: FL

New in version 0.9.8.

Attention

If the DFA is not minimal, the method loops forever

distR()[source]¶

Evaluate the right distinguishability language for a DFA

Return type: DFA

..seealso:: Cezar Câmpeanu, Nelma Moreira, Rogério Reis:: The distinguishability operation on regular languages. NCMA 2014: 85-100

distRMin()[source]¶

Compute distRMin for DFA

:rtype FL

..seealso:: Cezar Câmpeanu, Nelma Moreira, Rogério Reis:: The distinguishability operation on regular languages. NCMA 2014: 85-100

distTS()[source]¶

Evaluate the two-sided distinguishability language for a DFA

Return type: DFA

..seealso:: Cezar Câmpeanu, Nelma Moreira, Rogério Reis:: The distinguishability operation on regular languages. NCMA 2014: 85-100

dup()[source]¶

Duplicate the basic structure into a new DFA. Basically a copy.deep.

Return type: DFA

enumDFA(n=None)[source]¶

returns the set of words of words of length up to n accepted by self :param int n: highest length or all words if finite

Return type: list of strings or None

equal(other)[source]¶

Verify if the two automata are equivalent. Both are verified to be minimum and complete, and then one is matched against the other… Doesn’t destroy either dfa…

Parameters: other (DFA) – the other DFA
Return type: bool

evalSymbol(init, sym)[source]¶

Returns the state reached from given state through a given symbol.

Parameters

init (int) – set of current states indexes
sym (str) – symbol to be consumed

Returns

reached state

Return type

Raises

DFAsymbolUnknown – if symbol not in alphabet
DFAstopped – if transition function is not defined for the given input

evalSymbolI(init, sym)[source]¶

Returns the state reached from a given state.

Parameters

init (init) – current state
sym (str) – symbol to be consumed

Returns

reached state or -1

Return type

set of int

Raises

DFAsymbolUnknown – if symbol not in alphabet

New in version 0.9.5.

Note

this is to be used with non complete DFAs

evalSymbolL(ls, sym)[source]¶

Returns the set of states reached from a given set of states through a given symbol

Parameters

ls (set of int) – set of states indexes
sym (str) – symbol to be read

Returns

set of reached states

Return type

set of int

evalSymbolLI(ls, sym)[source]¶

Returns the set of states reached from a given set of states through a given symbol

Parameters

ls (set of int) – set of current states
sym (str) – symbol to be consumed

Returns

set of reached states

Return type

set of int

New in version 0.9.5.

Note

this is to be used with non complete DFAs

evalWord(wrd)[source]¶

Evaluates a word

Parameters: wrd (Word) – word
Returns: final state or None
Return type: int | None

New in version 1.3.3.

evalWordP(word, initial=None)[source]¶

Verifies if the DFA recognises a given word

Parameters

word (list of symbols.) – word to be recognised
initial (int) – starting state index

Return type

finalCompP(s)[source]¶

Verifies if there is a final state in strongly connected component containing s.

Parameters: s (int) – state
Returns: 1 if yes, 0 if no

hasTrapStateP()[source]¶

Tests if the automaton has a dead trap state

Return type: bool

New in version 1.1.

hyperMinimal(strict=False)[source]¶

Hyperminization of a minimal DFA

Parameters: strict (bool) – if strict=True it first minimizes the DFA
Returns: an hyperminimal DFA
Return type: DFA

See also

M. Holzer and A. Maletti, An nlogn Algorithm for Hyper-Minimizing a (Minimized) Deterministic Automata, TCS 411(38-39): 3404-3413 (2010)

Note

if strict=False minimality is assumed

inDegree(st)[source]¶

Returns the in-degree of a given state in an FA

Parameters: st (int) – index of the state
Return type: int

infix()[source]¶

Returns a dfa that recognizes infix(L(a))

Return type: DFA

initialComp()[source]¶

Evaluates the connected component starting at the initial state.

Returns: list of state indexes in the component
Return type: list of int

initialP(state)[source]¶

Tests if a state is initial

Parameters: state (int) – state index
Return type: bool

initialSet()[source]¶

The set of initial states

Returns: the set of the initial states
Return type: set of States

joinStates(lst)[source]¶

Merge a list of states.

Parameters: lst (iterable of state indexes.) – set of equivalent states

makeReversible()[source]¶

Make a DFA reversible (if possible)

See also

M.Holzer, s. Jakobi, M. Kutrib ‘Minimal Reversible Deterministic Finite Automata’

Return type: DFA

markNonEquivalent(s1, s2, data)[source]¶

Mark states with indexes s1 and s2 in given map as non equivalent states. If any back-effects exist, apply them.

Parameters

s1 (int) – one state’s index
s2 (int) – the other state’s index
data – the matrix relating s1 and s2

mergeStates(f, t)[source]¶

Merge the first given state into the second. If the first state is an initial state the second becomes the initial state.

Parameters

f (int) – index of state to be absorbed
t (int) – index of remaining state

Attention

It is up to the caller to remove the disconnected state. This can be achieved with `trim().

minimal(method='minimalHopcroft', complete=True)[source]¶

Evaluates the equivalent minimal complete DFA

Parameters

method – method to use in the minimization
complete (bool) – should the result be completed?

Returns

equivalent minimal DFA

Return type

minimalHopcroft()[source]¶

Evaluates the equivalent minimal complete DFA using Hopcroft algorithm

Returns: equivalent minimal DFA
Return type: DFA

See also

John Hopcroft,An n log{n} algorithm for minimizing states in a finite automaton.The Theory of Machines and Computations.AP. 1971

minimalHopcroftP()[source]¶

Tests if a DFA is minimal

Return type: bool

minimalIncremental(minimal_test=False)[source]¶

Minimizes the DFA with an incremental method using the Union-Find algorithm and memoized non-equivalence intermediate results

Parameters: minimal_test (bool) – starts by verifying that the automaton is not minimal?
Returns: equivalent minimal DFA
Return type: DFA

See also

M. Almeida and N. Moreira and and r. Reis.Incremental DFA minimisation. CIAA 2010. LNCS 6482. pp 39-48. 2010

minimalIncrementalP()[source]¶

Tests if a DFA is minimal

Return type: bool

minimalMoore()[source]¶

Evaluates the equivalent minimal automata with Moore’s algorithm

See also

John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, AW, 1979

Returns: minimal complete DFA
Return type: DFA

minimalMooreSq()[source]¶

Evaluates the equivalent minimal complete DFA using Moore’s (quadratic) algorithm

See also

John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, AW, 1979

Returns: equivalent minimal DFA
Return type: DFA

minimalMooreSqP()[source]¶

Tests if a DFA is minimal using the quadratic version of Moore’s algorithm

Return type: bool

minimalNCompleteP()[source]¶

Tests if a non necessarely complete DFA is minimal, i.e., if the DFA is non complete, if the minimal complete has only one more state.

Returns: True if not minimal
Return type: bool

Attention

obsolete: use minimalP

minimalNotEquivP()[source]¶

Tests if the DFA is minimal by computing the set of distinguishable (not equivalent) pairs of states

Return type: bool

minimalP(method='minimalMooreSq')[source]¶

Tests if the DFA is minimal

Parameters: method – the minimization algorithm to be used
Return type: bool

..note: if DFA non complete test if complete minimal has one more state

minimalWatson(test_only=False)[source]¶

Evaluates the equivalent minimal complete DFA using Waton’s incremental algorithm

Parameters: test_only (bool) – is it only to test minimality
Returns: equivalent minimal DFA
Return type: DFA
Raises: DFAnotComplete – if automaton is not complete

..attention::: automaton must be complete

minimalWatsonP()[source]¶

Tests if a DFA is minimal using Watson’s incremental algorithm

Return type: bool

notequal(other)[source]¶

Test non equivalence of two DFAs

Parameters: other (DFA) – the other DFA
Return type: bool

orderedStrConnComponents()[source]¶

Topological ordered list of strong components

New in version 1.3.3.

Return type: list

pairGraph()[source]¶

Returns pair graph

Return type: DiGraphVM

See also

A graph theoretic approach to automata minimality. Antonio Restivo and Roberto Vaglica. Theoretical Computer Science, 429 (2012) 282-291. doi:10.1016/j.tcs.2011.12.049 Theoretical Computer Science, 2012 vol. 429 (C) pp. 282-291. http://dx.doi.org/10.1016/j.tcs.2011.12.049

possibleToReverse()[source]¶: Tests if language is reversible

New in version 1.3.3.

pref()[source]¶

Returns a dfa that recognizes pref(L(self))

Return type: DFA

New in version 1.1.

print_data(data)[source]¶

Prints table of compatibility (in the context of the minimalization algorithm).

Parameters: data – data to print

product(other)[source]¶

Returns a DFA resulting of the simultaneous execution of two DFA. No final states set.

Note

this is a fast version of the method. The resulting state names are not meaningfull.

Parameters: other – the other DFA
Return type: DFA

productSlow(other, complete=True)[source]¶

Returns a DFA resulting of the simultaneous execution of two DFA. No final states set.

Note

this is a slow implementation for those that need meaningfull state names

New in version 1.3.3.

Parameters

other – the other DFA
complete (bool) – evaluate product as a complete DFA

Return type

reorder(dicti)[source]¶

Reorders states according to given dictionary. Given a dictionary (not necessarily complete)… reorders states accordingly.

Parameters: dicti (dict) – reorder dictionary

reverseTransitions(rev)[source]¶

Evaluate reverse transition function.

Parameters: rev (DFA) – DFA in which the reverse function will be stored

reversibleP()[source]¶

Test if an automaton is reversible

Return type: bool

sMonoid()[source]¶

Evaluation of the syntactic monoid of a DFA

Returns: the semigroup
Return type: SSemiGroup

sSemigroup()[source]¶

Evaluation of the syntactic semigroup of a DFA

Returns: the semigroup
Return type: SSemiGroup

shuffle(other, strict=False)[source]¶

CShuffle of two languages: L1 W L2

Parameters

other (DFA) – second automaton
strict (bool) – should the alphabets be necessary equal?

Return type

See also

C. Câmpeanu, K. Salomaa and s. Yu, Tight lower bound for the state complexity of CShuffle of regular languages. J. Autom. Lang. Comb. 7 (2002) 303–310.

simDiff(other)[source]¶

Symetrical difference

Parameters: other –
Returns

sop(other)[source]¶

Strange operation

Parameters: other (DFA) – the other automaton
Return type: DFA

See also

Nelma Moreira, Giovanni Pighizzini, and Rogério Reis. Universal disjunctive concatenation

and star. In Jeffrey Shallit and Alexander Okhotin, editors, Proceedings of the 17th Int. Workshop on Descriptional Complexity of Formal Systems (DCFS15), number 9118 in LNCS, pages 197–208. Springer, 2015.

New in version 1.2b2.

star(flag=False)[source]¶

Star of a DFA. If the DFA is not complete, it is completed.

..versionchanged: 0.9.6

Parameters: flag (bool) – plus instead of star
Returns: the result of the star
Return type: DFA

starI()[source]¶

Star of an incomplete DFA.

Returns: the Kleene closure DFA
Return type: DFA

stateChildren(state, strict=False)[source]¶

Set of children of a state

Parameters

strict (bool) – if not strict a state is never its own child even if a self loop is in place
state (int) – state id queried

Returns

map children -> multiplicity

Return type

dictionary

stronglyConnectedComponents()[source]¶

Dummy method that uses the NFA conterpart

New in version 1.3.3.

Return type: list

subword()[source]¶

Returns a dfa that recognizes subword(L(self))

Return type: dfa

New in version 1.1.

succintTransitions()[source]¶: Collects the transition information in a compact way suitable for graphical representation. :rtype: list of tupples

New in version 0.9.8.

suff()[source]¶

Returns a dfa that recognizes suff(L(self))

Return type: DFA

New in version 0.9.8.

syncPower()[source]¶

Evaluates the Power automata for the action of each symbol

Returns: The Power automata being the set of all states the initial state and all singleton states final.
Return type: DFA

toADFA()[source]¶

Try to convert DFA to ADFA

Returns: the same automaton as a ADFA
Return type: ADFA
Raises: notAcyclic – if this is not an acyclic DFA

New in version 1.2.

Changed in version 1.2.1.

toDFA()[source]¶

Dummy function. It is already a DFA

Returns: a self deep copy
Return type: DFA

toNFA()[source]¶

Migrates a DFA to a NFA as dup()

Returns: DFA seen as new NFA
Return type: NFA

transitions()[source]¶: Iterator over transitions :rtype: symbol, int

transitionsA()[source]¶: Iterator over transitions :rtype: symbol, int

uniqueRepr()[source]¶

Normalise unique string for the string icdfa’s representation. .. seealso:: TCS 387(2):93-102, 2007 https://www.dcc.fc.up.pt/~nam/publica/tcsamr06.pdf

Returns: normalised representation
Return type: list
Raises: DFAnotComplete – if DFA is not complete

universalP(minimal=False)[source]¶

Checks if the automaton is universal through minimisation

Parameters: minimal (bool) – is the automaton already minimal?
Return type: bool

unmark()[source]¶

Unmarked NFA that corresponds to a marked DFA: in which each alfabetic symbol is a tuple (symbol, index)

Returns: a NFA
Return type: NFA

usefulStates(initial_states=None)[source]¶

Set of states reacheable from the given initial state(s) that have a path to a final state.

Parameters: initial_states (iterable of int) – starting states
Returns: set of state indexes
Return type: set of int

static vDescription()[source]¶

Generation of Verso interface description

New in version 0.9.5.

Returns: the interface list

witness()[source]¶

Witness of non emptyness

Returns: word
Return type: str

witnessDiff(other)[source]¶

Returns a witness for the difference of two DFAs and:

0	if the witness belongs to the other language
1	if the witness belongs to the self language

Parameters: other (DFA) – the other DFA
Returns: a witness word
Return type: list of symbols
Raises: DFAequivalent – if automata are equivalent

NFA¶

class NFA[source]¶

Class for Non-deterministic Finite Automata (CEpsilon-transitions allowed).

Variables

States (list) – set of states.
sigma (set) – alphabet set.
Initial (set) – initial state indexes.
Final (set) – set of final states indexes.
delta (dict) – the transition function.

HKeqP(other, strict=True)[source]¶

Test NFA equivalence with extended Hopcroft-Karp method

See also

J. E. Hopcroft and r. M. Karp. A Linear Algorithm for Testing Equivalence of Finite Automata.TR 71–114. U. California. 1971

Parameters

other – NFA
strict – if True checks for same alphabets

Returns

Boolean

addEpsilonLoops()[source]¶: Add epsilon loops to every state :return: self

Attention

in-place modification

New in version 1.0.

addInitial(stateindex)[source]¶

Add a new state to the set of initial states.

Parameters: stateindex (int) – index of new initial state

addTransition(sti1, sym, sti2)[source]¶

Adds a new transition. Transition is from sti1 to sti2 consuming symbol sym. sti2 is a unique state, not a set of them.

Parameters

sti1 (int) – state index of departure
sti2 (int) – state index of arrival
str (str) – symbol consumed

addTransitionQ(srci, dest, symb, qfuture, qpast)[source]¶

Add transition to the new transducer instance.

Parameters

qpast (set) – past queue
qfuture (set) – future queue
symb – symbol
dest (int) – destination state
srci (int) – source state

New in version 1.0.

autobisimulation()[source]¶

Largest right invariant equivalence between states of the NFA

Returns: Incomplete equivalence relation (transitivity, and reflexivity not calculated) as a set of unordered pairs of states
Return type: Set of frozensets

See also

Ilie&Yu, 2003

autobisimulation2()[source]¶

Alternative space-efficient definition of NFA.autobisimulation.

Returns: Incomplete equivalence relation (reflexivity, symmetry, and transitivity not calculated) as a set of pairs of states
Return type: list of tuples

closeEpsilon(st)[source]¶

Add all non CEpsilon transitions from the states in the CEpsilon closure of given state to given state.

Parameters: st (int) – state index

computeFollowNames()[source]¶

Computes the follow set to use in names

Return type: list

concat(other, middle='middle')[source]¶

Concatenation of NFA

Parameters

middle (str) – glue state name
other (FA) – the other NFA

Returns

the result of the concatenation

Return type

NFA

countTransitions()[source]¶

Number of transitions of a NFA

Return type: int

delTransition(sti1, sym, sti2, _no_check=False)[source]¶

Remove a transition if existing and perform cleanup on the transition function’s internal data structure.

Parameters

sti1 (int) – state index of departure
sti2 (int) – state index of arrival
sym – symbol consumed
_no_check (bool) – dismiss secure code

Note

unused alphabet symbols will be discarded from sigma.

deleteStates(del_states)[source]¶

Delete given iterable collection of states from the automaton.

Parameters: del_states (set|list) – collection of int representing states

Note

delta function will always be rebuilt, regardless of whether the states list to remove is a suffix, or a sublist, of the automaton’s states list.

detSet(generic=False)[source]¶

Computes the determination uppon a followFromPosition result

Return type: NFA

deterministicP()[source]¶

Verify whether this NFA is actually deterministic

Return type: bool

dotFormat(size='20,20', filename=None, direction='LR', strict=False, maxlblsz=6, sep='\n') → str[source]¶

A dot representation

Parameters

direction (str) – direction of drawing - “LR” or “RL”
size (str) – size of image
filename (str) – output file name
sep (str) – line separator
maxlblsz (int) – max size of labels before getting removed
strict (bool) – use limitations of label sizes

Returns

the dot representation

Return type

New in version 0.9.6.

Changed in version 1.2.1.

dup()[source]¶

Duplicate the basic structure into a new NFA. Basically a copy.deep.

Return type: NFA

elimEpsilon()[source]¶

Eliminate CEpsilon-transitions from this automaton.

:rtype : NFA

Attention

performs in place modification of automaton

Changed in version 1.1.1.

eliminateEpsilonTransitions()[source]¶: Eliminates all epslilon-transitions with no state addition

Attention

in-place modification

eliminateTSymbol(symbol)[source]¶

Delete all trasitions through a given symbol

Parameters: symbol (str) – the symbol to be excluded from delta

Attention

in place alteration of the automata

New in version 0.9.6.

enumNFA(n=None)[source]¶

returns the set of words of words of length up to n accepted by self :param int n: highest lenght or all words if finite

Return type: list of strings or None

epsilonClosure(st)[source]¶

Returns the set of states CEpsilon-connected to from given state or set of states.

Parameters: st (int|set) – state index or set of state indexes
Returns: the list of state indexes CEpsilon connected to st
Return type: set of int

Attention

st must exist.

epsilonP()[source]¶

Whether this NFA has CEpsilon-transitions

Return type: bool

epsilonPaths(start, end)[source]¶

All states in all paths (DFS) through empty words from a given starting state to a given ending state.

Parameters

start (int) – start state
end (int) – end state

Returns

states in CEpsilon paths from start to end

Return type

set of states

equivReduced(equiv_classes)[source]¶

Equivalent NFA reduced according to given equivalence classes.

Parameters: equiv_classes (UnionFind) – Equivalence classes
Returns: Equivalent NFA
Return type: NFA

evalSymbol(stil, sym)[source]¶

Set of states reacheable from given states through given symbol and CEpsilon closure.

Parameters

stil (set|list) – set of current states
sym (str) – symbol to be consumed

Returns

set of reached state indexes

Return type

set

Raises

DFAsymbolUnknown – if symbol is not in alphabet

evalWordP(word)[source]¶

Verify if the NFA recognises given word.

Parameters: word (str) – word to be recognised
Return type: bool

finalCompP(s)[source]¶

Verify whether there is a final state in strongly connected component containing given state.

Parameters: s (int) – state index
Returns: :: bool

followFromPosition()[source]¶: computes follow automaton from a Position automaton :rtype: NFA

half()[source]¶: Half operation

New in version 0.9.6.

hasTransitionP(state, symbol=None, target=None)[source]¶

Whether there’s a transition from given state, optionally through given symbol, and optionally to a specific target.

Parameters

state (int) – source state
symbol (str) – optional transition symbol
target (int) – optional target state

Returns

if there is a transition

Return type

homogeneousFinalityP()[source]¶

Tests if states have incoming transitions froms states with different finalities

Return type: bool

homogenousP(x)[source]¶

Whether this NFA is homogenous; that is, for all states, whether all incoming transitions to that state are through the same symbol.

Parameters: x – dummy parameter to agree with the method in DFAr
Return type: bool

initialComp()[source]¶

Evaluate the connected component starting at the initial state.

Returns: list of state indexes in the component
Return type: list of int

lEquivNFA()[source]¶

Equivalent NFA obtained from merging equivalent states from autobisimulation of this NFA’s reversal.

Return type: NFA

Note

returns copy of self if autobisimulation renders no equivalent states.

lrEquivNFA()[source]¶

Equivalent NFA obtained from merging equivalent states from autobisimulation of this NFA, and from autobisimulation of its reversal; i.e., merges all states that are equivalent w.r.t. the largest right invariant and largest left invariant equivalence relations.

Return type: NFA

Note

returns copy of self if autobisimulations render no equivalent states.

minimal()[source]¶

Evaluates the equivalent minimal DFA

Returns: equivalent minimal DFA
Return type: DFA

minimalDFA()[source]¶

Evaluates the equivalent minimal complete DFA

Returns: equivalent minimal DFA
Return type: DFA

product(other)[source]¶

Returns a NFA (skeletom) resulting of the simultaneous execution of two DFA.

Parameters: other (NFA) – the other automata
Return type: NFA

Note

No final states are set.

Attention

the name EmptySet is used in a unique special state name
the method uses 3 internal functions for simplicity of code (really!)

rEquivNFA()[source]¶

Equivalent NFA obtained from merging equivalent states from autobisimulation of this NFA.

Return type: NFA

Note

returns copy of self if autobisimulation renders no equivalent states.

renameStatesFromPosition()[source]¶

Rename states of a Glushkov automaton using the positions of the marked RE

Return type: NFA

reorder(dicti)[source]¶

Reorder states indexes according to given dictionary.

Parameters: dicti (dict) – state name reorder

Note

dictionary does not have to be complete

reversal()[source]¶

Returns a NFA that recognizes the reversal of the language

Returns: NFA recognizing reversal language
Return type: NFA

reverseTransitions(rev)[source]¶

Evaluate reverse transition function.

Parameters: rev (NFA) – NFA in which the reverse function will be stored

setInitial(statelist)[source]¶

Sets the initial states of an NFA

Parameters: statelist (set|list|int) – an iterable of initial state indexes

shuffle(other)[source]¶

Shuffle of a NFA

Parameters: other (FA) – an FA

Returns: the resulting NFA: NFA: the resulting NFA

star(flag=False)[source]¶

Kleene star of a NFA

Parameters: flag (bool) – plus instead of star?
Returns: the resulting NFA
Return type: NFA

stateChildren(state, strict=False)[source]¶

Set of children of a state

Parameters

strict (bool) – if not strict a state is never its own child even if a self loop is in place
state (int) – state id queried

Returns

children states

Return type

Set of int

stronglyConnectedComponents()[source]¶

Strong components

Return type: list

New in version 1.0.

subword()[source]¶

returns a nfa that recognizes subword(L(self))

Return type: nfa

succintTransitions()[source]¶: Collects the transition information in a compact way suitable for graphical representation. :rtype: list

toDFA()[source]¶

Construct a DFA equivalent to this NFA, by the subset construction method.

Return type: DFA

Note

valid to CEpsilon-NFA

toNFA()[source]¶

Dummy identity function

Return type: NFA

toNFAr()[source]¶

NFA with the reverse mapping of the delta function.

Returns: shallow copy with reverse delta function added
Return type: NFAr

uniqueRepr()[source]¶: Dummy representation. Used DFA.uniqueRepr() :rtype: tuple

usefulStates(initial_states=None)[source]¶

Set of states reacheable from the given initial state(s) that have a path to a final state.

Parameters: initial_states (set of int or list of int) – set of initial states
Returns: set of state indexes
Return type: set of int

static vDescription()[source]¶

Generation of Verso interface description

Return type: list

New in version 0.9.5.

witness()[source]¶

Witness of non emptiness

Returns: word
Return type: str

wordImage(word, ist=None)[source]¶

Evaluates the set of states reached consuming given word

Parameters

word (list of stings) – the word
ist (int) – starting state index (or set of)

Returns

the set of ending states

Return type

Set of int

NFAr¶

class NFAr[source]¶

Class for Non-deterministic Finite Automata with reverse delta function added by construction.: Includes efficient methods for merging states.

addTransition(sti1, sym, sti2)[source]¶

Adds a new transition. Transition is from sti1 to sti2 consuming symbol sym. sti2 is a unique state, not a set of them. Reversed transition function is also computed

Parameters

sti1 (int) – state index of departure
sti2 (int) – state index of arrival
sym (str) – symbol consumed

delTransition(sti1, sym, sti2, _no_check=False)[source]¶

Remove a transition if existing and perform cleanup on the transition function’s internal data structure and in the reversal transition function

Parameters

sti1 (int) – state index of departure
sti2 (int) – state index of arrival
sym (str) – symbol consumed
_no_check (bool) – dismiss secure code

deleteStates(del_states)[source]¶

Delete given iterable collection of states from the automaton. Performe deletion in the transition function and its reversal.

Parameters: del_states (set or list of int) – collection of int representing states

elimEpsilonO()[source]¶

Eliminate CEpsilon-transitions from this automaton, with reduction of states through elimination of CEpsilon-cycles, and single CEpsilon-transition cases.

Returns: itself
Return type

Attention

performs inplace modification of automaton

homogenousP(inplace=False)[source]¶

Checks is the automaton is homogenous, i.e.the transitions that reaches a state have all the same label.

Parameters: inplace (bool) – if True performs CEpsilon transitions elimination
Returns: True if homogenous
Return type: bool

mergeStates(f, t)[source]¶

Merge the first given state into the second. If first state is an initial or final state, the second becomes respectively an initial or final state.

Parameters

f (int) – index of state to be absorbed
t (int) – index of remaining state

Attention

It is up to the caller to remove the disconnected state. This can be achieved with `trim().

mergeStatesSet(tomerge, target=None)[source]¶

Merge a set of states with a target merge state. If the states in the set have transitions among them, those transitions will be directly merged into the target state.

Parameters

tomerge (Set of int) – set of states to merge with target
target (int) – optional target state

Note

if target state is not given, the minimal index with be considered.

Attention

The states of the list will become unreacheable, but won’t be removed. It is up to the caller to remove them. That can be achieved with trim().

toNFA()[source]¶

Turn into an instance of NFA, and remove the reverse mapping of the delta function.

Returns: shallow copy without reverse delta function
Return type: NFA

unlinkSoleIncoming(state)[source]¶

If given state has only one incoming transition (indegree is one), and it’s through CEpsilon, then remove such transition and return the source state.

Parameters: state (int) – state to check
Returns: source state
Return type: int or None

Note

if conditions aren’t met, returned source state is None, and automaton remains unmodified.

unlinkSoleOutgoing(state)[source]¶

If given state has only one outgoing transition (outdegree is one), and it’s through CEpsilon, then remove such transition and return the target state.

Parameters: state (int) – state to check
Returns: target state
Return type: int or None

Note

if conditions aren’t met, returned target state is None, and automaton remains unmodified.

SSemiGroup¶

class SSemiGroup[source]¶

Class support for the Syntactic SemiGroup.

Variables

elements – list of tuples representing the transformations
words – a list of pairs (index of the prefix transformation, index of the suffix char)
gen – a list of the max index of each generation
sigma – set of symbols

WordI(i)[source]¶

Representative of an element given as index

Parameters: i (int) – index of the element
Returns: the first word originating the element
Return type: str

WordPS(pref, sym)[source]¶

Representative of an element given as prefix symb

Parameters

pref (int) – prefix index
sym (int) – symbol index

Returns

word

Return type

add(tr, pref, sym, tmpLists)[source]¶

Try to add a new transformation to the monoid

Parameters

tr (tuple of int) – transformation
pref (int or None) – prefix of the generating word
sym (int) – suffix symbol
tmpLists (pairs of lists as (elements,words)) – this generation lists

addGen(tmpLists)[source]¶

Add a new generation to the monoid

Parameters: tmpLists (pair of lists as (elements, words)) – the new generation data

EnumL¶

class EnumL(aut, store=False)[source]¶

Class for enumerate FA languages: See: Efficient enumeration of words in regular languages, M. Ackerman and J. Shallit, Theor. Comput. Sci. 410, 37, pp 3461-3470. 2009. http://dx.doi.org/10.1016/j.tcs.2009.03.018

Variables

aut (FA) – Automaton of the language
tmin (dict) – table for minimal words for each s in aut.States
Words (list) – list of words (if stored)
sigma (list) – alphabet

New in version 0.9.8.

enum(m)[source]¶

Enumerates the first m words of L(A) according to the lexicographic order if there are at least m words. Otherwise, enumerates all words accepted by A.

Parameters: m (int) – max number of words

enumCrossSection(n)[source]¶

Enumerates the nth cross-section of L(A)

Parameters: n (int) – nonnegative integer

abstract fillStack(w)[source]¶: Abstract method :param str w: :type w: str

iCompleteP(i, q)[source]¶

Tests if state q is i-complete

Parameters

i (int) – int
q (int) – state index

abstract initStack()[source]¶: Abstract method

minWord(m)[source]¶: Computes the minimal word of length m accepted by the automaton :param m: :type m: int

abstract minWordT(n)[source]¶: Abstract method :param int n: :type n: int

abstract nextWord(w)[source]¶: Abstract method :param w: :type w: str

Functions¶

saveToString¶

saveToString(aut, sep='&')[source]¶

Finite automata definition as a string using the input format.

New in version 0.9.5.

Changed in version 0.9.6: Names are now used instead of indexes.

Changed in version 0.9.7: New format with quotes and alphabet

Parameters

aut (FA) – the FA
sep (str) – separation between lines

Returns

the representation

Return type

stringToDFA¶

stringToDFA(s, f, n, k)[source]¶

Converts a string icdfa’s representation to dfa.

Parameters

s (list) – canonical string representation
f (list) – bit map of final states
n (int) – number of states
k (int) – number of symbols

Returns

a complete dfa with sigma [k], States [n]

Return type